Properties

Label 2-12274-1.1-c1-0-4
Degree $2$
Conductor $12274$
Sign $1$
Analytic cond. $98.0083$
Root an. cond. $9.89991$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 9-s + 6·11-s + 2·12-s − 2·13-s + 4·14-s + 16-s − 17-s − 18-s − 8·21-s − 6·22-s − 2·24-s − 5·25-s + 2·26-s − 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 34-s + 36-s + 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.74·21-s − 1.27·22-s − 0.408·24-s − 25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12274\)    =    \(2 \cdot 17 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(98.0083\)
Root analytic conductor: \(9.89991\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12274} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12274,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.710684355\)
\(L(\frac12)\) \(\approx\) \(1.710684355\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.49297642827790, −15.59701261222785, −15.39957496865513, −14.59229133657598, −14.20721122146874, −13.54751282503082, −13.06122787928628, −12.22246497714437, −11.86326758752552, −11.18175634901061, −10.19142998858016, −9.732669058602829, −9.365288500086130, −8.846226352027827, −8.369548906816165, −7.399864759355155, −7.098172718852221, −6.163391057224065, −5.995804647017067, −4.507091358111067, −3.767322815035053, −3.254464349437996, −2.528539852222390, −1.762334388164159, −0.6075611584413628, 0.6075611584413628, 1.762334388164159, 2.528539852222390, 3.254464349437996, 3.767322815035053, 4.507091358111067, 5.995804647017067, 6.163391057224065, 7.098172718852221, 7.399864759355155, 8.369548906816165, 8.846226352027827, 9.365288500086130, 9.732669058602829, 10.19142998858016, 11.18175634901061, 11.86326758752552, 12.22246497714437, 13.06122787928628, 13.54751282503082, 14.20721122146874, 14.59229133657598, 15.39957496865513, 15.59701261222785, 16.49297642827790

Graph of the $Z$-function along the critical line